American Journal of Epidemiology
ª The Author 2012. Published by Oxford University Press on behalf of the Johns Hopkins Bloomberg School of
Public Health. All rights reserved. For permissions, please e-mail: [email protected].
Vol. 175, No. 9
DOI: 10.1093/aje/kwr399
Advance Access publication:
March 29, 2012
Practice of Epidemiology
The Probability of Undetected Wild Poliovirus Circulation After Apparent Global
Interruption of Transmission
Dominika A. Kalkowska, Radboud J. Duintjer Tebbens, and Kimberly M. Thompson*
* Correspondence to Kimberly M. Thompson, Kid Risk, Inc., P.O. Box 590129, Newton, MA 02459 (email: [email protected]).
Initially submitted June 15, 2011; accepted for publication October 5, 2011.
The Global Polio Laboratory Network maintains active surveillance for circulating live polioviruses by obtaining
and testing stool samples from patients with acute flaccid paralysis. However, most poliovirus infections occur with
no symptoms, and questions remain about the probability of undetected wild poliovirus (WPV) circulation after the
apparent interruption of WPV transmission in different populations. In the context of making decisions about the
timing of oral poliovirus vaccine cessation following global eradication of WPV, policy-makers need an understanding of this probability as a function of time. Prior modeling of the probability of undetected circulation relied on
relatively simple models and assumptions, which limits extrapolation to current conditions. In this analysis, the
authors revisit the topic and highlight important considerations for policy-makers related to the impact of initial
conditions and seasonality and emphasize the need to focus on appropriate characterization of conditions in the
last likely reservoirs of the virus. The authors conclude that the probability of undetected WPV circulation may vary
significantly for different poliovirus serotypes, places, and conditions, which suggests that achieving the same level of
confidence about the true interruption of WPV transmission will require different periods of time for different situations.
disease outbreaks; disease transmission, infectious; models, statistical; poliomyelitis; poliovirus; risk assessment;
surveillance; vaccination
Abbreviations: CFP, case-free period; ECDF, empirical cumulative distribution function; IPV, inactivated poliovirus vaccine; OPV,
oral poliovirus vaccine; POE, probability of extinction; TSC, time of silent circulation; WPV, wild poliovirus.
As the Global Polio Eradication Initiative drives the number
of paralytic poliomyelitis cases caused by wild polioviruses
(WPVs) toward zero, questions arise about the possibility of
undetected or silent circulation of infection, particularly since
most poliovirus infections occur asymptomatically. The last
case of WPV type 2 occurred in northern India in 1999, and
ongoing, active surveillance for cases of acute flaccid paralysis reveals no evidence of undetected transmission or reemergence. Circulation of all WPVs has been successfully stopped
in 3 of the 6 World Health Organization regions, and endemic
circulation of WPV types 1 and 3 continues in shrinking geographic areas that export WPV and cause outbreaks. Typical
estimates suggest that permanent paralytic poliomyelitis
occurs at a rate of approximately 1 case per 200 infections,
on average, in immunologically naive (i.e., fully susceptible)
individuals, although the rates differ by serotype (1–3).
Two different polio vaccines provide effective protection
from paralytic poliomyelitis, which adds to the complexity of
modeling the transmission of poliovirus infections and population immunity (4). Inactivated poliovirus vaccine (IPV)
provides direct systemic immunity to vaccine recipients but
requires an injection, comes at a comparatively higher cost,
and does not provide significant enteric mucosal immunity
to prevent participation in fecal-oral transmission (4–6). The
easier-to-administer and less expensive oral poliovirus vaccine
(OPV) contains a live attenuated virus that infects the vaccine
recipient and induces both mucosal and systemic immunity,
and OPV infections can spread to others, providing secondary
immunity (2, 4, 6). However, OPV can cause sporadic cases
of vaccine-associated paralytic poliomyelitis in vaccine recipients or their contacts (approximately 1 in 2.5 million doses in
developed countries) (7–10), and it may lead to outbreaks of
936
Am J Epidemiol. 2012;175(9):936–949
Probability of Undetected Wild Poliovirus Circulation
937
N
W
b
pN
OPV Infectious
Incubating OPV
V
V11
V21
V12
V22
Permanently
and fully
immune
Fully susceptible
(1 – p)N
S
R
Incubating WPV
W
W11
WPV Infectious
W21
W12
W22
Cases
Rw
PIR
Figure 1. Schematic of a theoretical population according to poliovirus infection state (susceptible, infected, or recovered) and movements
between states for the oral poliovirus vaccine (OPV) model. Solid arrows indicate flows, and dashed arrows demonstrate the ways in which inputs
included in the sensitivity analyses influence parts of the model. (N, initial size of the total population (200,000); S, number of susceptible
individuals; W1j, number of individuals incubating a wild poliovirus (WPV) infection according to a 2-step process, j ¼ 1, 2; W2j, number of
WPV-infected/infectious individuals according to a 2-step process, j ¼ 1, 2; V1j, number of individuals incubating an OPV infection according to
a 2-step process, j ¼ 1, 2; V2j, number of OPV-infected/infectious individuals according to a 2-step process, j ¼ 1, 2; l, death rate ¼ 1/life
expectancy (1/45 [1/years] ¼ 1/16,425 [1/days]); a, population growth rate (2% [%/year] ¼ 2/365% [1/days])); m, per capita birth rate ¼ population
growth rate þ death rate [1/days]; p, fraction of newborns vaccinated successfully at birth (60% for OPV and 80% for inactivated poliovirus vaccine);
d, transition rate of incubation distribution ¼ 1/(one-half the duration of the incubation period) (1/(7 days/2) ¼ 1/3.5 [1/days]); c, transition rate of
infectivity distribution ¼ 1/(one-half the duration of infectivity) (1/(30 days/2) ¼ 1/15 [1/days]); Rw, basic reproduction number for WPV; b, rate of
sufficiently close contacts for transmission of WPV ([Rw(d þ l)2(c þ l)2)/(d2(2c þ l)]); kW, force of infection for WPV (b (W21 þ W22)/N); b, relative
infectivity of the vaccine virus (OPV) compared with that of WPV (25%); kV, force of infection for OPV (b kW); PIR, paralysis-to-infection ratio (1/200)).
vaccine-derived poliovirus infection (11–15). Ultimately,
eradicating all poliomyelitis cases will require OPV cessation and stopping all transmission of vaccine-derived
polioviruses (16–21). Coordinated OPV cessation should
occur only after assurance that WPV has stopped circulating
everywhere (22, 23).
Surveillance conducted by the Global Polio Laboratory
Network and vaccination response following detection of
outbreaks represent critical components of the Global Polio
Eradication Initiative (16–18, 24, 25). The surveillance system targets the collection of 2 stool samples from patients
with acute flaccid paralysis, recognized as paralysis characterized by rapid progression of loss of muscle tone or weak,
floppy (i.e., not spastic or rigid) muscle with loss of voluntary movement. Cases of acute flaccid paralysis occur from
nonpoliovirus causes in approximately 1 per 100,000 children
younger than 15 years of age, and the Global Polio Laboratory
Network confirms a poliomyelitis case following WPV isolation (24, 25). The surveillance system performs well, as
demonstrated by the robustness of its results, which have
supported certification of 3 World Health Organization regions as free of indigenous WPVs (26–28) and the absence
of any subsequent reported indigenous WPVs in these reAm J Epidemiol. 2012;175(9):936–949
gions. The World Health Organization used the experience
of the Pan American Health Organization as a basis for
requiring a period of at least 3 years of no paralytic polio
cases detected by acute flaccid paralysis surveillance for
certification (23). Debanne and Rowland (29) reported less
than a 5% chance of undetected indigenous WPV circulation after 4 years since the last reported confirmed case,
based on their statistical analysis of Pan American Health
Organization data. However, regions and areas within them
differ, and indigenous WPV strains reemerged after years of
no detected clinical cases in a few places that failed to stop
transmission and experienced some significant surveillance
gaps (30, 31).
Eichner and Dietz (32) characterized the probability of
undetected poliovirus circulation using a relatively simple
dynamic and stochastic infection transmission model of a
hypothetical, homogeneously mixed population of 200,000
people in a relatively low-income country that grows exponentially at a constant annual rate (a) (e.g., 2% growth in the
base case). In related deterministic, dynamic transmission
models, Eichner and Hadeler (33) calculated and compared
the theoretical thresholds for the vaccination coverage required
to stop transmission of polioviruses for a hypothetical
938 Kalkowska et al.
N
Incubating WPV
Partially susceptible
(1 – )pN
c
S
WPV Infectious
W
12
11
21
22
Permanently
and fully
immune
R
Fully susceptible
(1 – p)N
Incubating WPV
W
S
W11
WPV Infectious
W12
W21
W22
Cases
Rw
PIR
Figure 2. Schematic of a theoretical population according to poliovirus infection state (susceptible, infected, or recovered) and movements
e number
between states for the inactivated poliovirus vaccine (IPV) model. (See Figure 1 for symbols that appear on both Figure 1 and Figure 2. S,
e 1j, number of partially susceptible individuals incubating a wild poliovirus (WPV)
of partially susceptible individuals (after IPV vaccination); W
e 2j, number of individuals infected/infectious with a WPV infection
infection according to a 2-step process, j ¼ 1, 2 (as W1j for fully susceptibles); W
according to a 2-step process, j ¼ 1, 2 (as W2j for fully susceptibles); ec, transition rate of infectivity distribution of partially susceptible individuals with
an infectious period reduced to 20% ¼ l/(20% times one-half the duration of infectivity of IPV-vaccinated individuals) (1/(0.2 3 (30 days/2)) ¼ 1/3 [1/
days]); ae, percentage of IPV vaccine recipients who derive full protection from infection following successful IPV vaccination (30%); ce, relative
susceptibility of successfully IPV-vaccinated individuals who remain partially susceptible (50%); kW, force of infection for WPV (b (W21 þ W22 þ
e 21 þ W
e 22)/N ); PIR, paralysis-to-infection ratio).
W
population, and Eichner et al. (34) explored issues related
to characterizing the minimum population size required for
poliovirus persistence. Eichner and Dietz (32) initialized their
simulation at the WPV endemic equilibrium, at which point
they started vaccination of newborns according to an effective
vaccination coverage percentage (p). They estimated the
probability of extinction (POE) in the population by counting the number of iterations that led to the disruption of
transmission (i.e., zero prevalence of WPV-infected individuals in the population) and dividing this by the total number
of iterations in the simulation. Extinction does not occur in
all iterations in a simulation, and the POE depends on the
selected model input values.
Eichner and Dietz (32) assumed that a case occurred at
time 0, used a time horizon of 10 years, and recorded the
time since the last paralytic case on a monthly basis (i.e., 120
simulated month observations), which they called a case-free
period (CFP). They characterized the probability of WPV
circulation persisting given a CFP of length t months, which
they called the ‘‘probability of silent infections,’’ by dividing
the number of all CFPs of length t months with WPV present
by the total number of all CFPs of length t months. Intuitively,
short CFPs will often occur while WPV still circulates, leading to a high fraction of CFPs with circulation, but as the CFP
increases, the probability of continued WPV circulation decreases. Using these methods, Eichner and Dietz (32) reported
that not observing a case for 3 years provided 95% confidence
of local extinction of WPV infections in their simple hypothetical model. We use the notation CFPx% to indicate the
time at which fewer than x% of the CFP values occur with
WPV present, and we note that Eichner and Dietz (32) presented both CFP5% and CFP1% values, with full recognition
that the selection of the level of confidence represents a policy
choice.
The possibility of silent transmission continues to raise
concerns about outbreaks, potential reestablishment of widespread transmission, and the level of confidence in achieving
actual WPV eradication. Given renewed interest in assessing
the probability of undetected WPV circulation after global
WPV eradication, we reconstructed Eichner and Dietz’s
model (32) to explore the importance of framing choices they
made about initial conditions and seasonality and discuss
limitations associated with using simple models to support
policy choices.
MATERIALS AND METHODS
Starting with the differential equations, notation, and input
values provided by Eichner and Dietz (32), we reconstructed
their OPV and IPV models in Java using the Eclipse open
development platform (Eclipse Foundation, Inc., Ottawa,
Ontario, Canada). Figures 1 and 2 provide a graphical
Am J Epidemiol. 2012;175(9):936–949
Probability of Undetected Wild Poliovirus Circulation
A)
10
9
8
Stochastic Iteration
7
6
5
4
3
2
1
0
B)
0
1
2
3 4 5 6 7 8
Time in Simulation, years
9
10
0
1
2
3 4 5 6 7 8
Time in Simulation, years
9
10
10
9
8
Stochastic Iteration
7
6
5
4
3
2
1
0
Figure 3. Occurrence of simulated cases of paralytic poliomyelitis
for 10 randomly selected stochastic iterations using A) oral poliovirus
vaccine or B) inactivated poliovirus vaccine and starting at the endemic equilibrium.
representation of the OPV and IPV models, respectively.
Equivalent to the set of differential equations provided by
Eichner and Dietz (32), these schematics show the assumed
Am J Epidemiol. 2012;175(9):936–949
939
infection states, processes relevant for OPV (Figure 1) and
IPV (Figure 2) vaccination, and transitions affected by the
various model inputs. The models assume that incubation
(i.e., the state between virus exposure and becoming infectious) and infection (i.e., the state of infectiousness to
others) each occur in 2 stages, with half of the total duration
assigned to each stage. The models assume that only those
individuals who have no prior poliovirus infection and are not
effectively vaccinated with IPV can develop paralytic poliomyelitis according to the paralysis-to-infection ratio (e.g.,
1:200 for the base case). As Figures 1 and 2 show, the models
further assume that individuals remain fully and permanently
protected from reinfection by any poliovirus following recovery from a WPV or OPV infection. This represents a very
important simplification, because individuals with prior exposure to live or inactivated polioviruses can get reinfected
at any age and potentially participate in infection transmission, albeit with a lower probability of becoming infected
and with shorter durations of infection and infectiousness
than fully susceptible individuals (4, 35–38).
For both the OPV and IPV models, Eichner and Dietz (32)
explored the sensitivity of the CFP5% (in years) to different
values of the basic reproduction number of WPV (Rw), initial
population size (N), annual population growth (a), paralysisto-infection ratio, life expectancy (1/death rate ¼ 1/l), and
effective vaccination coverage (p). For their OPV model, they
also varied the ratio of the reproduction numbers of OPV to
WPV (b), and for their IPV model they varied the percentage
of IPV recipients who derive full protection from infection
following successful IPV vaccination (~
a) and the relative
susceptibility (compared with fully susceptible individuals) of
successfully IPV-vaccinated individuals who remain partially
susceptible (~
c).
We explored the impact of model framing assumptions
related to the initial conditions, seasonality, time horizon,
and recording interval used in the simulation for CFPs. We
explored the impact of changing the initial conditions from
the WPV endemic equilibrium (32) to the equilibrium with
55% effective vaccination coverage, which required that
we calculate the new equilibrium state for the population
(see Appendix). We conducted the simulations starting at the
new equilibrium state of vaccination with 55% coverage for
both models and then increased vaccination to the effective
vaccination coverage levels used by Eichner and Dietz (32)
in their original 1-way sensitivity analyses. We explored
the impact of seasonality by varying the basic reproduction
number while maintaining a fixed average (Rw ¼ 12) using 2
different approaches: 1) a sine function with amplitude of
0.5 3 average Rw and the high season peaking on July 15
and 2) high and low seasons (with varied lengths) with the
low season Rw fixed (RwL ¼ 8) and RwH varied to keep the
overall Rw ¼ 12 and maintaining the high season centered in
time around July 15. We found unstable results with only
1,000 iterations, so we used 50,000 iterations for all of our
simulations to obtain stable results within 60.1 year.
Although it is not observable in the field, in the context of
modeling we can characterize the time of silent circulation
(TSC) as the time between the last case and the end of WPV
infection transmission for each iteration of the simulation
that ends with extinction. After performing 50,000 iterations
940 Kalkowska et al.
Table 1. Independent Reanalysis of the 1-Way Sensitivity Analyses Performed by Eichner and Dietz (32) for POE, CFP5%,a and CFP1% and
Values of TSC5%b and TSC1% for 50,000 Iterations After Introducing Oral Poliovirus Vaccine Into a Hypothetical Population of 200,000 People
With Wild Poliovirus Circulating at the Endemic Equilibrium
Model Input Varied
Lowest Value
Low Value
Base Case Value
High Value
Higher Value
1/1,000
1/400
1/200
1/100
1/50
87.6
87.6
87.6
87.6
87.6
CFP5%, years
5.1
3.7
2.8
2.1
1.5
CFP1%, years
7.1
5.1
3.8
2.9
2.1
TSC5%, years
3.6
2.9
2.3
1.8
1.3
TSC1%, years
4.8
3.9
3.2
2.4
Paralysis-to-infection ratio
POE, %
Effective vaccination coverage (p), %
Highest Value
1.8
50
55
60
65
14.2
49.3
87.6
99.8
CFP5%, years
3.0
3.0
2.8
2.3
2.0
1.3
CFP1%, years
3.5
3.7
3.8
3.1
2.5
1.6
TSC5%, years
2.1
2.2
2.3
2.2
1.9
1.3
TSC1%, years
2.6
2.9
3.2
2.9
2.5
1.6
POE, %
Basic reproduction no. for WPV (Rw)
95
100
100
10
12
14
16
99.0
95.1
87.6
80.5
73.3
CFP5%, years
2.6
2.7
2.8
2.9
2.9
CFP1%, years
3.5
3.7
3.8
3.9
3.8
TSC5%, years
2.4
2.4
2.3
2.3
2.2
TSC1%, years
3.3
3.3
3.2
3.1
POE, %
Infectivity of OPV relative to WPV (b), %
8
70
2.9
15
20
25
30
41.3
64.4
87.6
99.1
CFP5%, years
3.0
3.0
2.8
2.5
2.1
1.5
CFP1%, years
3.7
3.9
3.8
3.4
2.8
1.9
TSC5%, years
2.1
2.3
2.3
2.3
2.1
1.5
TSC1%, years
2.8
3.0
3.2
3.1
2.8
100,000
200,000
300,000
400,000
1,000,000
POE, %
Initial size of the total population (N )
POE, %
35
55
100
100
1.9
96.4
87.6
81.2
74.4
45.9
CFP5%, years
2.7
2.8
2.9
3.0
3.1
CFP1%, years
3.7
3.8
3.9
4.0
4.0
TSC5%, years
2.3
2.3
2.4
2.4
2.4
TSC1%, years
3.1
3.2
3.2
3.2
3.1
Table continues
and creating the empirical cumulative distribution function
(ECDF) of all of the TSC values (n 50,000), we can
estimate the time after which the probability of occurrence
of a longer TSC becomes less than x% (TSCx%). Thus, the
TSC5% represents the time t that satisfies the equation 1 ECDF(t) ¼ P[TSC > t] 0.05. Although the Global Polio
Laboratory Network cannot actually identify a case as the
last one at the time it occurs and can only identify a case as
last after having observed no cases for a long time, the TSC
represents a quantity of interest because it tells us how much
time we can expect to pass between the last paralytic case and
actual extinction of infections. In contrast, by considering the
fraction of CFPs that occur in the context of WPV circulation,
the CFPx% metric addresses the probability that silent WPV
circulation continues as a function of time without observed
cases.
RESULTS
Figure 3 depicts the occurrence of simulated paralytic
cases (dots) as a function of time for 10 randomly selected
stochastic iterations. Cases occur frequently at the beginning
of the simulation and decrease in frequency over time, which
makes it easy to see how CFP values near 0 months arise.
Cases occur frequently just after the start of the model (i.e.,
when conditions begin to change from the endemic equilibrium due to the initiation of vaccination of some newborns),
which leads to many CFP values of 0 months, because when
checking monthly the ‘‘case-free’’ time elapsed between cases
that occur in the same month or in sequential months equals 0.
As the cases become less frequent, the simulation may record multiple CFP values between cases. Thus, in a given
stochastic iteration, if a CFP value of 6 months gets recorded,
Am J Epidemiol. 2012;175(9):936–949
Probability of Undetected Wild Poliovirus Circulation
941
Table 1. Continued
Model Input Varied
Annual population growth rate (a), %
Lowest Value
Low Value
Base Case Value
High Value
0
1
2
3
98.6
95.0
87.6
79.3
CFP5%, years
2.7
2.7
2.8
2.9
CFP1%, years
3.7
3.8
3.8
3.9
TSC5%, years
2.4
2.4
2.3
2.3
TSC1%, years
3.3
3.3
3.2
POE, %
Life expectancy (1/l), years
POE, %
45
50
85.9
87.6
89.8
2.9
2.8
2.8
CFP1%, years
3.9
3.8
3.8
TSC5%, years
2.3
2.3
2.3
3.2
TSC1%, years
3.2
3.2
3
5
7
9
92.2
90.2
87.6
86.2
CFP5%, years
2.4
2.6
2.8
3.0
CFP1%, years
3.3
3.6
3.8
4.1
TSC5%, years
2.1
2.2
2.3
2.5
TSC1%, years
2.6
3.0
3.2
POE, %
Duration of infectivity period (c), days
POE, %
Highest Value
3.1
40
CFP5%, years
Duration of incubation period (d), days
Higher Value
3.3
20
25
30
35
40
95.9
92.4
87.6
84.0
78.6
CFP5%, years
2.0
2.4
2.8
3.2
3.7
CFP1%, years
2.8
3.3
3.8
4.5
4.9
TSC5%, years
1.8
2.0
2.3
2.6
2.9
TSC1%, years
2.4
2.8
3.2
3.5
3.8
Abbreviations: CFP, case-free period; OPV, oral poliovirus vaccine; POE, probability of extinction; TSC, time of silent circulation; WPV, wild
poliovirus.
a
CFPx %, case-free period after which the probability of WPV circulation becomes less than x %.
b
TSCx %, time at which the probability of silent WPV circulation after the true last case becomes less than x %.
implying at least 6 months between 2 cases or after the last
case, then this must follow recorded CFP values of 5, 4, 3, 2,
and 1 months. Most of the iterations in Figure 3 clearly lead
to extinction of WPV, as shown by the black lines that begin
at the time of the last case and end when WPV circulation
stops in the model. However, several of the iterations show
cases followed by a time period when WPV continues to
circulate until the end of the 10-year time horizon, as shown
by the dashed black lines in Figure 3. We found that shortening the time horizon to 5 years significantly lowered the
POE (i.e., from 87.6% to 40.5%) and decreased the average
of the CFPs by not allowing for any CFP observations of
more than 4.1 years, but we confirmed that the time horizon of 10 years used in the original analysis (32) remains
sufficient. We found that changing the recording interval
leads to a negligible impact (results not shown).
Tables 1 and 2 show our results from repeating the 1-way
sensitivity analyses reported by Eichner and Dietz (32) for
vaccination with OPV and IPV, respectively, including the
POE, CFP5%, CFP1%, TSC5%, and TSC1% estimates.
Tables 1 and 2 include several values and model inputs
not reported by Eichner and Dietz (32) that we felt provided
Am J Epidemiol. 2012;175(9):936–949
important insights. Our base case results match the CFP5%
values reported by Eichner and Dietz (32) within 0.1 years
(i.e., 2.8 vs. 2.9 using OPV; 3.0 vs. 2.9 using IPV) and the
POE values within 2% (i.e., 87.6% vs. 87.2% using OPV;
77.1% vs. 75.1% using IPV). The differences in our results
reflect our use of the original equations presented by Eichner
and Dietz (32), which allow mortality to occur from all states
shown in Figures 1 and 2, and from the greater stability of our
results given our use of 50,000 iterations for all simulations.
Increasing the paralysis-to-infection ratio significantly
changes the length of time required to observe cases, with
lower values implying the need to wait longer because cases
occur less frequently. Changing the paralysis-to-infection
ratio does not, however, affect the transmission of infection
or the POE; it merely changes the frequency with which we
can observe evidence of circulation as cases. We expanded
the range to include a ratio of 1:1,000, which may represent
a more appropriate paralysis-to-infection ratio for WPV
type 3 (3), and the results suggested the need to potentially
consider different policies for the 3 poliovirus serotypes.
We observe consistent changes in the POE values with
increases in different input values. Increases in the effective
942 Kalkowska et al.
Table 2. Independent Reanalysis of the 1-Way Sensitivity Analyses Performed by Eichner and Dietz (32) for POE, CFP5%,a and CFP1% and
Values of TSC5%b and TSC1% for 50,000 Iterations After Introducing Inactivated Poliovirus Vaccine Into a Hypothetical Population of 200,000
People With Wild Poliovirus Circulating at the Endemic Equilibrium
Model Input Varied
Lowest Value
Low Value
Base Case Value
High Value
Higher Value
1/1,000
1/400
1/200
1/100
1/50
77.1
77.1
77.1
77.1
77.1
CFP5%, years
5.2
3.9
3.0
2.2
1.6
CFP1%, years
6.1
4.8
3.8
2.9
2.1
TSC5%, years
3.4
2.8
2.3
1.8
1.4
TSC1%, years
4.2
3.5
3.0
2.4
Paralysis-to-infection ratio
POE, %
Effective vaccination coverage (p), %
1.8
70
75
80
85
90
21.3
48.5
77.1
95.0
99.9
CFP5%, years
3.0
3.0
3.0
2.6
2.2
CFP1%, years
3.5
3.6
3.8
3.7
3.0
TSC5%, years
2.0
2.1
2.3
2.3
2.1
TSC1%, years
2.6
2.7
3.0
3.0
2.8
POE, %
Basic reproduction no. for WPV (Rw)
10
12
14
16
94.2
84.4
77.1
70.4
63.9
CFP5%, years
2.8
2.9
3.0
3.0
3.0
CFP1%, years
3.8
3.9
3.8
3.7
3.6
TSC5%, years
2.4
2.3
2.3
2.2
2.1
TSC1%, years
3.3
3.1
3.0
2.8
100,000
200,000
300,000
400,000
1,000,000
90.4
77.1
63.8
53.2
18.9
CFP5%, years
2.8
3.0
3.0
3.1
3.1
CFP1%, years
3.8
3.8
3.8
3.8
3.8
TSC5%, years
2.2
2.3
2.3
2.3
2.2
TSC1%, years
3.0
3.0
2.9
2.9
2.8
0
1
2
3
POE, %
8
Highest Value
Initial size of the total population (N )
POE, %
Annual population growth rate (a), %
POE, %
93.9
85.4
77.1
66.3
CFP5%, years
2.8
2.9
3.0
3.0
CFP1%, years
3.9
3.9
3.9
3.7
TSC5%, years
2.4
2.4
2.3
2.2
TSC1%, years
3.3
3.2
3.0
2.8
2.7
Table continues
vaccination coverage (p), the relative transmissibility of
OPV compared with WPV (b), life expectancy (1/l), and
the percentage of successfully IPV-vaccinated individuals
who derive full protection (~
a) increase the POE. In contrast,
increases in population size (N), population growth rate
(a), basic reproduction number (RW), relative susceptibility for IPV-vaccinated individuals who remain partially
susceptible to asymptomatic infections (~
c), duration of
incubation period (d), and duration of infectivity (c) decrease the POE. Although we might expect that the CFPx%
and TSCx% would directly track the behavior of the POE, we
see from Tables 1 and 2 that this does not always occur. Most
of the 1-way sensitivity results show essentially no impact on
the results rounded to the total number of years, and we focus
on the results that show a difference of at least 6 months
between the minimum and maximum values (i.e., d, c, p,
and b). Intuitively, increases in d and c increase the time
between successive infections in a chain of transmission
and thus between successive cases, which proportionally
decrease POE and increase CFPx% and TSCx% values.
We see a general decline of CFPx% and TSCx% values as
p and b increase, although in some cases we see the
CFPx% and TSCx% values increase slightly before they
decline. As derived by Eichner and Hadeler (33), p, RW, b,
and a~ all relate to virus transmission and affect the threshold vaccination coverage (p*) required for extinction:
1
1
Rw
:
p*OPV ¼ 1
ð1bÞ and p*IPV ¼ 1
Rw
Rw Rw ð1~
aÞR1
Theoretically, using a deterministic model, for p 69%
for OPV and p 98% for IPV, WPV extinction occurs with
Am J Epidemiol. 2012;175(9):936–949
Probability of Undetected Wild Poliovirus Circulation
943
Table 2. Continued
Model Input Varied
Lowest Value
Life expectancy (1/l), years
POE, %
Low Value
Base Case Value
High Value
40
45
50
74.5
77.1
77.7
3.0
3.0
3.0
CFP5%, years
CFP1%, years
3.8
3.8
3.9
TSC5%, years
2.2
2.3
2.3
TSC1%, years
Percentage of successful IPV
vaccinees who derive full
protection (ã)
POE, %
2.9
3.0
Higher Value
3.0
10
20
30
40
50
67.9
72.3
77.1
80.1
83.9
CFP5%, years
3.0
3.0
3.0
2.9
2.8
CFP1%, years
3.8
3.8
3.8
3.8
3.8
TSC5%, years
2.2
2.2
2.3
2.3
2.3
TSC1%, years
2.9
Relative susceptibility for IPV-vaccinated
individuals who remain partially
susceptible to asymptomatic
infection (c~), %
POE, %
CFP5%, years
0
3.0
3.0
3.0
3.0
25
50
75
100
96.8
88.7
77.1
62.8
46.8
2.5
2.8
3.0
3.0
3.1
CFP1%, years
3.5
3.8
3.8
3.8
3.6
TSC5%, years
2.2
2.3
2.3
2.2
2.1
TSC1%, years
3.0
3.1
3.0
2.8
2.7
3
5
7
9
85.0
81.3
77.1
71.9
CFP5%, years
2.6
2.8
3.0
3.2
CFP1%, years
3.4
3.6
3.8
4.1
TSC5%, years
2.0
2.1
2.3
2.4
TSC1%, years
2.7
2.8
3.0
3.1
Duration of incubation period (d), days
POE, %
Duration of infectivity period (c), days
20
25
30
35
40
90.2
84.4
77.1
68.0
60.9
CFP5%, years
2.2
2.6
3.0
3.4
3.7
CFP1%, years
3.0
3.4
3.8
4.2
4.6
TSC5%, years
1.7
2.0
2.3
2.5
2.8
TSC1%, years
2.3
2.7
3.0
3.3
3.6
POE, %
Highest Value
Abbreviations: CFP, case-free period; IPV, inactivated poliovirus vaccine; POE, probability of extinction; TSC, time of silent circulation; WPV,
wild poliovirus.
a
CFPx %, case-free period after which the probability of WPV circulation becomes less than x %.
b
TSCx %, time at which the probability of silent WPV circulation after the true last case becomes less than x %.
certainty (33). However, in stochastic models, extinction can
occur due to chance in some iterations for values of p < p*,
and it may not occur in all iterations for p > p* within the
time horizon, which implies a fuzzier concept than a fixed
threshold. Nonetheless, the threshold equations provide helpful insights by showing that increasing b decreases p*OPV,
which explains the nonmonotonic changes in the CFPx%
and TSCx% values in Tables 1 and 2. For values of p much
greater than p*, extinction typically occurs relatively quickly
in most iterations, which leads to relatively small CFPx%
and TSCx% values because the high population immunity
Am J Epidemiol. 2012;175(9):936–949
stops transmission. In contrast, when p approaches p*, we
observe that circulation can continue longer, because effective
vaccination coverage just over the threshold allows the number of susceptibles to build up relatively quickly, which
supports sustained transmission with long delays between
cases. For p values far below p*, cases occur frequently unless
extinction occurs by chance following an outbreak, leading to
a shorter CFPx% and only a short period of silent circulation
after the last case. Although 1-way sensitivity analyses do
not capture the multiway effects, investing in more extensive sensitivity analyses for this simple model did not make
944 Kalkowska et al.
Table 3. Impact of Different Initial Conditions and Effective Vaccination Coverage on POE, CFP5%,a and TSC5%b
Effective Vaccination Coverage (p), %
Oral Poliovirus Vaccine
55
60
65
Inactivated Poliovirus Vaccine
70
95
100
100
70
75
80
85
90
Start at endemic equilibrium
POE, %
49.2
87.6
99.7
21.8
49.0
77.1
95.1
99.8
CFP5%, years
3.1
2.8
2.3
2.0
1.4
3.0
3.1
3.0
2.6
2.2
TSC5%, years
2.3
2.4
2.3
1.9
1.4
2.0
2.2
2.3
2.3
2.2
POE, %
2.6
22.6
87.5
99.9
0.9
7.7
25.6
59.1
93.8
CFP5%, years
2.9
3.3
3.3
2.7
1.7
3.1
3.1
3.2
3.3
2.9
TSC5%, years
2.1
2.3
2.7
2.6
1.7
2.1
2.1
2.3
2.4
2.5
Start at 55% effective vaccination
coverage equilibrium
100
Abbreviations: CFP, case-free period; POE, probability of extinction; TSC, time of silent circulation.
CFP5%, case-free period after which the probability of WPV circulation becomes less than 5%.
b
TSC5%, time at which the probability of silent WPV circulation after the true last case becomes less than 5%.
a
A)
30
25
Rw
20
15
10
5
0
0
30
60
90
120
150
180
210
240
270
300
330
360
Time, days
C)
100
100
90
90
Probability of Silent Infections, %
Probability of Silent Infections, %
B)
80
70
60
50
40
30
20
80
70
60
50
40
30
20
10
10
0
0
0
1
2
3
Case-Free Period, years
4
5
0
1
2
3
Case-Free Period, years
4
5
Figure 4. Assumptions about the seasonality of poliovirus transmission (part A) and their impact on CFP5% for B) oral poliovirus vaccine and
C) inactivated poliovirus vaccine for the base case settings starting at the endemic equilibrium. —, constant; —, sine; – – –, H3-L9; - - - -, H6-L6;
– –, H9-L3. H and L represent the duration (in months) of the high season and the low season, respectively. (CFP5%, the time at which fewer than
5% of the case-free period (CFP) values occur with wild poliovirus present).
Am J Epidemiol. 2012;175(9):936–949
Probability of Undetected Wild Poliovirus Circulation
945
Table 4. Impact of Seasonality on POE, CFP5%,a and TSC5%b for Different High Season Lengths, by Type of Vaccine Introduced at the Start of
the Simulation
Oral Poliovirus Vaccine
Inactivated Poliovirus Vaccine
Seasonal Pattern
POE, %
CFP5%, years
TSC5%, years
POE, %
CFP5%, years
TSC5%, years
Base case
87.6
2.8
2.4
77.1
3.0
2.3
Sine function
91.4
2.6
2.3
78.8
2.8
2.2
H3-L9c
100.0
1.9
1.8
99.9
2.1
1.8
H4-L8
99.9
2.0
1.9
99.6
2.1
1.9
H5-L7
99.6
2.2
2.1
97.9
2.3
2.0
H6-L6
98.4
2.6
2.2
93.7
2.7
2.1
H7-L5
94.3
2.8
2.3
81.6
2.9
2.2
H8-L4
86.8
2.9
2.4
64.3
3.0
2.2
H9-L3
73.8
3.0
2.5
45.7
3.0
2.3
Abbreviations: CFP, case-free period; H, high season; L, low season; POE, probability of extinction; TSC, time of silent circulation.
CFP5%, case-free period after which the probability of wild poliovirus circulation becomes less than 5%.
b
TSC5%, time at which the probability of silent wild poliovirus circulation after the true last case becomes less than 5%.
c
H3-L9 indicates 3 months of high season (H3) and 9 months of low season (L9).
a
sense because the model does not capture complexity that
arises in real situations or the real correlations that exist
between inputs. Thus, we concluded that future analyses
will need to focus on appropriately characterizing the inputs
relevant to the conditions in the last likely WPV reservoirs
using more sophisticated models and ensure the validity of key
framing assumptions.
Table 3 shows the impact of changing the different initial
conditions. Starting the simulation at equilibrium with 55%
effective vaccination coverage instead of the endemic
equilibrium decreases the POE significantly. For example,
the column with OPV base case (effective vaccine coverage
equal to 60%) shows that the POE drops from 87.6% to
22.6% with the change in the initial conditions, which demonstrates that the different dynamics of building up population immunity matter (39, 40). Specifically, the greater force
of infection from WPV at the endemic equilibrium compared
with OPV vaccination creates a rapid drop in population
immunity when vaccination starts, which leads to a greater
probability of reaching zero prevalence of WPV by chance.
Notably, for the OPV base case (p ¼ 60%), starting at the
equilibrium with 55% effective vaccination coverage instead of the endemic equilibrium increases the CFP5% from
2.8 years to 3.3 years, with similar changes for the IPV base
case. The 55% effective vaccination coverage falls far below
the theoretical threshold for extinction with OPV or IPV (i.e.,
69% or 98%) (33), and introducing vaccination decreases the
spread of WPV, which lengthens the time between cases (32).
Consequently, observing a longer CFP provides relatively
less assurance of disruption of WPV circulation when starting
at the new equilibrium, which implies the need to wait longer
to achieve high confidence about the actual cessation of WPV
circulation.
In addition to observing a big impact from the selection of
the initial conditions, we found that seasonality may also play
an important role. Figure 4A shows our different assumptions
about seasonality, which produce the same overall average
Am J Epidemiol. 2012;175(9):936–949
Rw. Parts B and C of Figure 4 show how the CFP5% curves
shift as a function of seasonality when introducing OPV and
IPV, respectively, for the base case values starting at the
endemic equilibrium. Table 4 summarizes the changes in
the POE, CFP5%, and TSC5%, including results with additional high-low season divisions beyond the three depicted
in Figure 4. Our analysis suggests that a short, intense high
season followed by a relatively long low season (i.e., H3-L9)
increases the POE, which changes from 87.6% to 100.0%
using OPV (from 77.1% to 99.9% using IPV) in comparison
with the base case. Longer high seasons tend to lengthen the
time that WPV circulation continues after the last paralytic
case, as expected due to the monotonicity between length of
the high season and TSC5%.
DISCUSSION
Although the current certification criterion (23) appears
reasonable in the context of the existing model results, our
analyses provide additional context and raise several issues
for consideration. First, serotype differences in the paralysisto-infection ratios may imply significant differences between
the relevant CFP and TSC values. Second, we highlight the
importance of our observation of a wide range of POE results. Concerns about silent circulation of WPVs depend on
first achieving the goal of apparent WPVextinction. Some of
the combinations of inputs and conditions we explored demonstrate the importance of using strategies beyond low levels
of routine immunization of newborns to achieve extinction.
In the context of implementing the Global Polio Eradication
Initiative, much of the success with respect to achieving
relatively high levels of population immunity in some
countries depended on conducting supplemental immunization activities. This suggests that future modeling efforts
will need to deal with a much more complex set of assumptions about the underlying population immunity and the
946 Kalkowska et al.
actual interventions used to achieve global WPV eradication (4).
Third, our results suggest that framing assumptions about
the initial conditions and seasonality may represent important factors that simple sensitivity analyses focused on varying inputs will miss. The endemic equilibrium represents the
starting point that provides the maximum population immunity from WPV infections. However, countries typically do not
instantly increase coverage to a sufficiently high level to
eradicate; instead, they gradually increase coverage. In the
context of the Global Polio Eradication Initiative, countries
might further increase coverage and add supplemental immunization activities to interrupt transmission (41, 42). Our
results suggest that starting at more realistic conditions with
respect to population immunity may imply the need to wait
longer to certify a population as polio-free in some places. We
anticipate that the actual path taken to pursue WPV eradication, the policies following the last detected case, and accurate
representation of the actual field conditions will influence
the probability of undetected WPV circulation over time, and
this implies the need to move away from reliance on simple,
hypothetical models toward models of the actual conditions
that exist in the last real WPV reservoirs. Our results also
suggest that policy-makers need to evaluate the specific
concepts modeled (CFP, TSC, and POE) and consider the
information provided by complimentary metrics.
Fourth, the choice of the confidence level that policy-makers
require for certification comes with real trade-offs. The risks
associated with continued use of OPV create incentives to
stop OPV use soon after the certification of global WPV
eradication, but some uncertainty will remain about the possibility of undetected WPV circulation and about potential
risks of reintroduction (unintentional or intentional). While
we did not focus on the length of time that circulating vaccinederived polioviruses might silently circulate, we note that
although they behave similarly to WPVs in some respects,
additional modeling will need to specifically address the
issue of potential silent circulating vaccine-derived poliovirus
transmission in the context of any real populations of concern
with appropriate consideration of the actual conditions.
Several limitations in this and the prior analyses (32–34)
warrant additional consideration. The models ignore the real
heterogeneity that exists in the world and the actual conditions
that affect the ability of poliovirus to be transmitted in populations and subgroups. In addition, while successful vaccination with either OPV or IPV probably leads to permanent
immunity from paralytic poliomyelitis, vaccinated individuals
can still become reinfected if challenged with a live poliovirus
(6) and can participate in the transmission of infection. Individuals with only IPV-induced immunity can become infected nearly as frequently as susceptibles and can excrete
as much or nearly as much virus in feces as susceptibles
(5). This implies the need to modify the model framing assumption that a fraction of IPV vaccinees become fully immune to infection and to include the effects of reinfection and
waning (4, 42). Actual effective vaccination coverage varies
for an individual country as a function of time and among
subgroups within the population (43), and nonrandom, heterogeneous mixing occurs. The models also implicitly assume
perfect surveillance, although surveillance quality may vary
significantly, with poor-quality surveillance often coinciding
with poverty, low vaccine coverage, and other conditions that
favor poliovirus transmission. In future studies, investigators
will need to address the impacts of imperfect surveillance
quality, because lower-quality surveillance will increase the
CFP and TSC, and they should also consider potential opportunities to enhance and improve surveillance, perhaps including environmental surveillance (44).
We hope this study provides additional insights about the
potential of undetected WPV circulation and that future
research will address the key limitations, which should go
beyond consideration of uncertainty in the model inputs and
focus specifically on the complex population immunity that
exists in the real populations likely to represent the last WPV
reservoirs.
ACKNOWLEDGMENTS
Author affiliations: Kid Risk, Inc., Newton, Massachusetts
(Dominika A. Kalkowska, Radboud J. Duintjer Tebbens,
Kimberly M. Thompson); and Delft Institute of Applied
Mathematics, Delft University of Technology, Delft, the
Netherlands (Dominika A. Kalkowska).
This analysis was supported by a grant from the World
Health Organization Polio Research Committee.
The authors thank the World Health Organization Polio
Research Committee for supplying funding. They thank Prof.
Dr. Martin Eichner for helpful discussions and for sharing
his computer code, which facilitated comparisons with
these results. The authors also thank Drs. Mark Pallansch,
Steve Wassilak, and Steve Cochi for helpful comments.
The findings and conclusions in this report are those of
the authors and do not necessarily represent the views of the
World Health Organization.
Conflict of interest: none declared.
REFERENCES
1. Bernier RH. Some observations on poliomyelitis lameness
surveys. Rev Infect Dis. 1984;6(suppl 2):S371–S375.
2. Sutter RW, Kew OM, Cochi SL. Poliovirus vaccine (live).
In: Plotkin SA, Orenstein WA, Offit PA, eds. Vaccines. 5th ed.
Philadelphia, PA: WB Saunders/Elsevier; 2008:631–686.
3. Nathanson N, Kew OM. From emergence to eradication: the
epidemiology of poliomyelitis deconstructed. Am J Epidemiol.
2010;172(11):1213–1229.
4. Duintjer Tebbens RJ, Pallansch MA, Kew OM, et al. A dynamic
model of poliomyelitis outbreaks: learning from the past to help
inform the future. Am J Epidemiol. 2005;162(4):358–372.
5. Cuba IPV Study Collaborative Group. Randomized, placebocontrolled trial of inactivated poliovirus vaccine in Cuba.
N Engl J Med. 2007;356(15):1536–1544.
6. Onorato IM, Modlin JF, McBean AM, et al. Mucosal immunity
induced by enhance-potency inactivated and oral polio vaccines.
J Infect Dis. 1991;163(1):1–6.
7. Alexander LN, Seward JF, Santibanez TA, et al. Vaccine policy
changes and epidemiology of poliomyelitis in the United States.
JAMA. 2004;292(14):1696–1701.
Am J Epidemiol. 2012;175(9):936–949
Probability of Undetected Wild Poliovirus Circulation
8. Andrus JK, Strebel PM, de Quadros CA, et al. Risk of vaccineassociated paralytic poliomyelitis in Latin America, 1989–91.
Bull World Health Organ. 1995;73(1):33–40.
9. Kohler KA, Banerjee K, Gary Hlady W, et al. Vaccineassociated paralytic poliomyelitis in India during 1999:
decreased risk despite massive use of oral polio vaccine.
Bull World Health Organ. 2002;80(3):210–216.
10. Esteves K. Safety of oral poliomyelitis vaccine: results of
a WHO enquiry. Bull World Health Organ. 1988;66(6):739–746.
11. Estı́variz CF, Watkins MA, Handoko D, et al. A large vaccinederived poliovirus outbreak on Madura Island—Indonesia,
2005. J Infect Dis. 2008;197(3):347–354.
12. Kew O, Morris-Glasgow V, Landaverde M, et al. Outbreak of
poliomyelitis in Hispaniola associated with circulating type 1
vaccine-derived poliovirus. Science. 2002;296(5566):356–359.
13. Kew OM, Wright PF, Agol VI, et al. Circulating vaccinederived polioviruses: current state of knowledge. Bull World
Health Organ. 2004;82(1):16–23.
14. Yang CF, Naguib T, Yang SJ, et al. Circulation of endemic type
2 vaccine-derived poliovirus in Egypt from 1983 to 1993.
J Virol. 2003;77(15):8366–8377.
15. Kew OM, Sutter RW, de Gourville EM, et al. Vaccine-derived
polioviruses and the endgame strategy for global polio eradication. Annu Rev Microbiol. 2005;59:587–635.
16. Thompson KM, Tebbens RJ, Pallansch MA, et al. The risks,
costs, and benefits of possible future global policies for managing
polioviruses. Am J Public Health. 2008;98(7):1322–1330.
17. World Health Organization. Global Polio Eradication
Initiative—Strategic Plan 2010–2012. Geneva, Switzerland:
World Health Organization; 2010.
18. Tebbens RJ, Pallansch MA, Kew OM, et al. Risks of paralytic
disease due to wild or vaccine-derived poliovirus after eradication. Risk Anal. 2006;26(6):1471–1505.
19. Thompson KM, Duintjer Tebbens RJ. The case for cooperation
in managing and maintaining the end of poliomyelitis: stockpile
needs and coordinated OPV cessation. Medscape J Med. 2008;
10(8):190.
20. Aylward RB, Sutter RW, Heymann DL. OPV cessation—the
final step to a ‘‘polio-free’’ world. Science. 2005;310(5748):
625–626.
21. World Health Organization. Cessation of Routine Oral Polio
Vaccine (OPV) Use After Global Polio Eradication: Framework for National Policy Makers in OPV-using Countries.
Geneva, Switzerland: World Health Organization; 2005.
22. Smith J, Leke R, Adams A, et al. Certification of polio eradication: process and lessons learned. Bull World Health Organ.
2004;82(1):24–30.
23. World Health Organization. Global Polio Eradication Initiative:
Strategic Plan 2004–2008. Geneva, Switzerland: World Health
Organization; 2004.
24. Hull BP, Dowdle WR. Poliovirus surveillance: building the
global polio laboratory network. J Infect Dis. 1997;175(suppl 1):
S113–S116.
25. World Health Organization. Polio Laboratory Manual. 4th ed.
Geneva, Switzerland: World Health Organization; 2004.
(http://www.who.int/immunization_monitoring/laboratory_
polio_resources/en/index.html). (Accessed May 14, 2011).
26. International Commission for the Certification of Poliomyelitis
Eradication. Third Meeting of the International Commission for
the Certification of Poliomyelitis Eradication (ICCPE). Final
Report—Conclusions and Recommendations. Washington, DC,
22–25 August 1994. Washington, DC: Pan American Health
Organization; 1994.
27. Regional Commission for the Certification of Poliomyelitis in
the Western Pacific Region, World Health Organization. Report.
Am J Epidemiol. 2012;175(9):936–949
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
947
Sixth Meeting of the Regional Commission for the Certification
of Poliomyelitis in the Western Pacific Region. Kyoto, Japan,
27–28 October 2000. Geneva, Switzerland: World Health
Organization; 2000. (http://www.wpro.who.int/internet/
resources.ashx/EPI/docs/Meetings/RCC/MTGRPT_RCC6.pdf).
(Accessed August 11, 2011).
European Regional Commission for the Certification of
Poliomyelitis Eradication, World Health Organization. Fifteenth
Meeting of the European Regional Commission for the Certification of Poliomyelitis Eradication. Copenhagen, Denmark,
19–21 June 2002. Geneva, Switzerland: World Health Organization; 2005. (http://www.euro.who.int/__data/assets/pdf_file/
0003/79374/E88105.pdf). (Accessed August 11, 2011).
Debanne SM, Rowland DY. Statistical certification of eradication of poliomyelitis in the Americas. Math Biosci. 1998;150(1):
83–103.
Progress toward poliomyelitis eradication—poliomyelitis
outbreak in Sudan, 2004. MMWR Morb Mortal Wkly Rep. 2005;
54(4):97–99.
Laboratory surveillance for wild and vaccine-derived polioviruses, January 2007–June 2008. Wkly Epidemiol Rec. 2008;
83(36):321–338.
Eichner M, Dietz K. Eradication of poliomyelitis: when can
one be sure that polio virus transmission has been terminated?
Am J Epidemiol. 1996;143(8):816–822.
Eichner M, Hadeler KP. Deterministic models for the eradication of poliomyelitis: vaccination with the inactivated (IPV)
and attenuated (OPV) polio virus vaccine. Math Biosci. 1995;
127(2):149–166.
Eichner M, Hadeler KP, Dietz K. Stochastic models for the
eradication of poliomyelitis: minimum population size for
polio virus persistence. In: Isham V, Medley GF, eds. Models
for Infectious Human Diseases: Their Structure and Relation
to Data. New York, NY: Cambridge University Press; 1996:
315–327.
Abbink F, Buisman AM, Doornbos G, et al. Poliovirus-specific
memory immunity in seronegative elderly people does not
protect against virus excretion. J Infect Dis. 2005;191(6):
990–999.
Goffe A, Pollock TM, Shand FL. Vaccination of adults with
a British oral poliomyelitis vaccine prepared from Sabin
strains. Br Med J. 1961;2(5247):272–275.
Kogon A, Spigland I, Elveback LR, et al. Oral poliovirus
vaccination in middle income families. Am J Epidemiol. 1965;
82(1):14–26.
Vaccine Administration Subcommittee, Japan Live Poliovaccine Research Commission. Evaluation of Sabin live
poliovirus vaccine in Japan. II. Clinical, virologic and
immunologic effects of vaccine in children. Jpn J Med Sci
Biol. 1966;19(6):277–291.
Sabin AB. Strategy for rapid elimination and continuing control of poliomyelitis and other vaccine preventable diseases
of children in developing countries. Br Med J (Clin Res Ed).
1986;292(6519):531–533.
Sabin AB, Ramos-Alvarez M, Alvarez-Amezquita J, et al.
Live, orally given poliovirus vaccine. Effects of rapid mass
immunization on population under conditions of massive
enteric infection with other viruses. JAMA. 1960;173(14):
1521–1526.
Hull HF, Ward NA, Hull BP, et al. Paralytic poliomyelitis:
seasoned strategies, disappearing disease. Lancet. 1994;
343(8909):1331–1337.
Duintjer Tebbens RJ, Pallansch MA, Cochi SL, et al. Economic
analysis of the Global Polio Eradication Initiative. Vaccine.
2010;29(2):334–343.
948 Kalkowska et al.
43. Thompson KM, Wallace GS, Duintjer Tebbens RJ, et al.
Trends in the risk of U.S. polio outbreaks and poliovirus
vaccine availability for response. Public Health Rep. 2012;
127(1):23–37.
44. Department of Vaccines and Biologicals, World Health Organization. Guidelines for Environmental Surveillance of Poliovirus Circulation. (Report no. WHO/V&B/03.03). Geneva,
Switzerland: World Health Organization; 2003.
APPENDIX
Derivation of New Initial Conditions for Equilibrium with
Vaccination
Using the notation developed by Eichner et al. (32–34),
we used Maple (MapleSoft, Waterloo, Ontario, Canada) to
derive the following equations for the new equilibrium with
oral poliovirus (OPV) vaccination:
NðtÞ ¼ Nð0ÞeðvlÞt
S
1
¼
N Rw
W11
¼
N
v
l
vp
1
Rw 1b d þ l
W12
¼
N
l
vp
d
v
Rw 1b ðd þ lÞ2
W21
¼
N
l
vp
d2
v
Rw 1b ðd þ lÞ2 ðc þ lÞ
W22
¼
N
l
vp
d2 c
v
Rw 1b ðd þ lÞ2 ðc þ lÞ2
V11
p
v
¼
1b dþl
N
V12
p
vd
¼
1 b ðd þ lÞ2
N
V21
p
vd2
¼
1 b ðd þ lÞ2 ðc þ lÞ
N
V22
p
vd2 c
¼
1 b ðd þ lÞ2 ðc þ lÞ2
N
Rw ¼
bd2 2c þ l
ðd þ lÞ2 ðc þ lÞ2
where Rw represents the basic reproduction number for wild
poliovirus (WPV) infection, Rv represents the basic reproduction number for vaccine virus infection, N equals the population size, l equals the death rate (i.e., 1/life expectancy),
m represents the per capita birth rate (population growth rate þ
death rate), d gives the transition rate of incubation distribution (one-half the duration of the incubation period), c represents the transition rate of infectivity distribution (one-half
the duration of infectivity), b provides the rate of transmission
of wild virus, b equals the relative infectivity of the vaccine
virus versus WPV, and p represents the effective vaccination
rate (i.e., the fraction of successfully vaccinated newborns).
For the inactivated poliovirus vaccine (IPV) model, we could
not solve the equations exactly using Maple, so we approximated the new equilibrium using the following equations:
NðtÞ ¼ Nð0ÞeðvlÞt
S
¼s
N
W11
1
v
¼
pð1 a~ÞR1
Rw 1~
c
N
v
c~ð1pÞ
þ ð1pÞRw l
dþl
s
W12
1
v
¼
pð1 a~ÞR1
Rw 1~
c
N
vd
c~ð1pÞ
þ ð1pÞRw l
s
ðd þ lÞ2
W21
1
v
¼
pð1 a~ÞR1
Rw 1~
c
N
vd2
c~ð1pÞ
þ ð1pÞRw l
s
ðd þ lÞ2 ðc þ lÞ
W22
1
v
¼
pð1 a~ÞR1
Rw 1~
c
N
vd2 l
c~ð1pÞ
þ ð1pÞRw l
s
ðd þ lÞ2 ðc þ lÞ2
S~ 1 sRw
¼
N
R1
e11
W
1
v
c~ ð1pÞ
¼
þ pð1 a~Þ
Rw s
dþl
1~
c R1
N
e12
1
vd
c~ ð1pÞ
W
¼
þ pð1 a~Þ
Rw s
1~
c R1
N
ðd þ lÞ2
Rv ¼ bRw ;
Am J Epidemiol. 2012;175(9):936–949
Probability of Undetected Wild Poliovirus Circulation
e21
W
1
vd2
c~ ð1pÞ
þ pð1 a~Þ
¼
Rw s
1~
c R1
N
ðd þ lÞ2 ð~
c þ lÞ
e22
W
1
vd2 ~c
c~ ð1pÞ
þ pð1 a~Þ
¼
Rw s
1~
c R1
N
ðd þ lÞ2 ð~c þ lÞ2
2
Rw ¼
bd 2c þ l
R1 ¼
þ ð~
cd2 vbðp~
aðc þ lÞ2 ð2~c þ lÞ ð~c þ lÞ2 ð2c þ lÞ
ðc ~cÞpðlc þ 2c~c þ l~cÞÞ þ lð~c þ lÞ2
3 ðd þ lÞ2 ðc þ lÞ2 ð1~
cÞÞs þ c~vð~c þ lÞ2
c~bd 2~
cþl
2
ðd þ lÞ ð~
c þ lÞ
individuals who remain partially susceptible to asymptomatic infection, and ~c gives the transition rate of infectivity for
partially susceptible individuals (i.e., one-half the duration
of infectivity of IPV-vaccinated individuals). We use s to
represent the susceptible fraction in the population, and
we approximate s by solving for the root of the following
equation (ignoring higher-order terms):
cÞs2
0 ¼ lbd2 ð2c þ lÞð~c þ lÞ2 ð1~
ðd þ lÞ2 ðc þ lÞ2
2
949
2
;
where R1 represents the reproduction number for a population with reduced susceptibility, a~ represents the percentage
of individuals vaccinated with IPV who derive full protection, c~ gives the relative susceptibility for IPV-vaccinated
Am J Epidemiol. 2012;175(9):936–949
3 ðd þ lÞ2 ðc þ lÞ2 ð1 pÞ:
We used Vensim (Ventana Systems, Inc., Harvard,
Massachusetts) to verify the equations for the initial conditions with 55% effective vaccination coverage in a separate,
deterministic implementation of the IPV model.